Prediction model of loess immersion settlement based on improved K-G model

Abstract

Loess is a problematic type of soil with a worldwide distribution due to its collapsibility. The temporal discontinuity and spatial nonuniformity of its collapsibility can bring severe damage to building foundations, roads and water pipelines. In this study, the relationship between the saturation and K-G model parameters is established based on indoor compression tests and collapsible tests; the deformation characteristics of loess immersed in water are studied via a large-scale trial immersion pit test. The test site is a circular pit with a diameter of 10 m. The loess is immersed for 46 days; the variation in its accumulated settlement over time is recorded for 60 days, and its deformation process is simulated using a self-designed programme. Results show that for the stress–strain relationship of unsaturated loess, the relationship between equivalent suction and saturation can be obtained through the principle of deformation equivalence and fitted using the exponential function. The maximum vertical displacements calculated in the simulation and on-site immersion pit experiment are 0.036 m and 0.032 m, respectively. Such relatively good consistency indicates that the proposed method can reasonably predict the collapse behaviour of loess due to immersion. This research provides a reliable method for the numerical simulation of loess immersion deformation, and the parameters in the model only need to be determined by conventional experiments.

Introduction

Loess is a terrestrial sediment that has been deposited and formed in arid and semi-arid regions (33–47° N) since the quaternary period. This type of soil is widely distributed in Asia, Europe, North America and South America and covers 13 × 10⁶ km² or approximately one-tenth of the global land area (Li et al. 2020). The distribution area of loess in China has reached 63 × 10⁴ km², especially in its Loess Plateau region. Loess is famous for its broad coverage, heavy thickness and continuous distribution (Zhu et al. 2022a). Its unique structure is characterised by the random distribution of large pores and vertical crack development due to its unique sedimentary environment and material composition. Under natural water content, but when immersed in water, loess becomes sensitive to collapse, which increases its settlement and reduces its load capacity (Feng et al. 2021; Meng et al. 2021). Figure 1a shows the crack formation on a building caused by an uneven foundation settlement due to water immersion. Figure 1b shows the ground subsidence caused by long-term rainfall. Figure 1c shows a collapse caused by the rupture of an underground water pipe. The strong water absorption of loess can lead to the collapse of large-scale structures. Therefore, studying the collapsibility of loess and understanding its laws are of great significance.

Fig. 1

Engineering disasters caused by loess collapse: a Crack formation on a building in Wuhan Donghu New Technology Development Zone; b road collapse on Shanfeng Road, Jinshazhou, Baiyun District, Guangzhou; and c rupture of an underground water pipeline at the intersection of Kunshan Road and Nujiang Street, Huanggu District, Shenyang

The collapse of loess is mainly determined by its structure, cementation and hydraulic properties (Ma et al. 2017; Zhang et al. 2019; Jian et al. 2022). The structural characteristics of loess mainly refer to its trellis pores. Loess particles are mainly composed of silt, accounting for over 50% of the soil. The clay particles in loess are cemented into aggregates or attached to the surface of sand and silt, and the silt and aggregates together form the skeleton of the supporting structure. Due to a relatively loose arrangement, few contact points can be found in loess (Li et al. 2019; Mokritskaya et al. 2019; Xiao et al. 2022). The cement of loess is formed by the gradual concentration and precipitation of soluble salts under arid or semi-dry climate conditions. The cementation material enhances the ability of soil particles to resist sliding and the self-weight compaction of the loess (Wang et al. 2020). Hydro-physical properties refer to the dissolution of some soluble salts in loess after immersion, which softens the cementitious material and reduces the strength of the skeleton (Ye et al. 2020; Zuo et al. 2020).

Previous studies on loess collapsibility have mainly conducted indoor experiments to explore the mechanism of loess collapsibility and to test mathematical models for quantitative description. Intact loess is retrieved from exploratory wells, and its collapsibility is tested via uniaxial or triaxial compression tests. This same approach has been widely used in practical engineering (Jia et al. 2020; Lai et al. 2020; Pan et al. 2021). However, the self-weight collapse deformation, collapse type and foundation collapse deformation of a site as measured via indoor tests often significantly differ from those measured via on-site tests mainly for three reasons. Firstly, these studies have a small sample size, and the original structure of the loess may be damaged during the sampling and preparation processes. Secondly, the pore distribution and loess skeleton structure of the loess sample taken within a specific burial depth range cannot represent the corresponding soil properties of the soil layer. Thirdly, the compression test limits the lateral deformation of the sample and the shear effect. However, shallow loess undergoes significant lateral deformation during the field immersion test, thereby increasing the vertical settlement deformation (Jiang et al. 2012; Wang et al. 2014). The field immersion test can directly measure the amount of collapsible deformation of the foundation (especially self-weight collapsibility) and the spatiotemporal distribution of collapsible deformation. Whilst this test can accurately and reliably determine the type of collapsible deformation of a site, its application is both expensive and time-consuming (Deng et al. 2018; Zhu et al. 2022b).

The construction scale and thickness of the loess layer in construction sites in China’s loess areas are increasing, and collapsible loess deformation has always been a problem for the engineering community. Given that collapsible deformation involves the relationship between stress state and water content and has abrupt and discontinuous characteristics, accurately simulating its development law can be challenging. Some scholars have built constitutive models for calculating loess collapsible deformation by taking the cause of collapsibility as the starting point, using matrix suction and saturation as the model parameters and combining them with the theory of unsaturated soil deformation (Yuan and Wang 2009; Xing et al. 2019). Meanwhile, other scholars have followed the principles of generalised plastic mechanics and considered the hydraulic hysteresis soil–water characteristic curve affected by deformation, established an incremental constitutive relationship for collapsible deformation and built a dual potential surface constitutive model for unsaturated soil (Zhang et al. 2020; Zhao et al. 2022; Hong et al. 2023). Some have also established a viscoelastic plastic model related to unsaturated soil rate from the perspectives of viscoelasticity and plasticity whilst considering the influence of strain rate on deformation and strength (Xu et al. 2017). With these long-term efforts of hydraulic and geotechnical engineers, people have gained a deeper understanding of the mechanism of loess collapsible deformation. However, most of the constitutive models available in the literature are developed based on conventional test results and consist of complex forms/matrices and numerous parameters. Specialised research that focuses on foundation instability and deformation caused by flooding in loess areas or strictly follows the consolidation theory of unsaturated soil to build a calculation model is yet to be conducted.

The nonlinear elasticity of soil is mainly studied using two models, namely, the E-µ and K-G constitutive models. In the K-G model, the mechanical concept is clear, the parameters are easy to measure and each parameter can be directly related to soil shear or bulk modulus, thereby making this model convenient for geotechnical engineering calculations (Cheng et al. 2018). To accurately understand the law of collapsible loess deformation and solve the uneven deformation caused by rainfall, this paper analyses and summarises reasonable constitutive models related to saturation, establishes a practical nonlinear model of unsaturated loess that introduces saturation and proposes a constitutive model and calculation method for simulating the wetting deformation of loess.

Methodology

Material preparation and characterisation

The loess used in the experiment is taken from Donglintuo Village, Yuncheng City, Shanxi Province at depths of 2 m and 4 m below ground level. The collected soil samples are cut into 20 to 30 cm³ on site, marked with up and down arrows and wrapped in plastic film and tape before being transported back to the laboratory. After their retrieval, the structure of the soil samples is checked to see any disturbances and to confirm if their quality meets the test requirements.

Experimental methods

Compression test and collapsibility test are two commonly used testing methods in the field of soil mechanics to study the deformation characteristics of soil. The compression test is conducted to study the compressional deformation characteristics of soil under external loads. This test helps understand the compression behaviour of soil under different pressures and the properties related to stress–strain relationships. The compression test aims to study the deformation characteristics of soil during the processes of moisture absorption and drying-induced shrinkage. This test helps understand the swelling and shrinkage behaviour of soil under different humidity conditions and potential volume changes.

Test instruments

The testing apparatus has been modified from an existing unsaturated triaxial apparatus, with the primary addition being the incorporation of an axial stress control mechanism and an immersion device. This setup facilitates permeability tests under various stress conditions, as well as triaxial shear tests along different stress paths. The principle of the apparatus is depicted in Fig. 2. The measurement precision achieved by the apparatus includes a volumetric change and drainage volume sensitivity of 0.01 cm³, axial deformation sensitivity of 0.01 mm and pore water pressure sensitivity of 1 kPa.

Fig. 2

Instrument of collapsibility test

Compression test

Multiple sets of specimens with initial water contents of 14%, 18% and 22% are prepared for the compression test. 14%, 18% and 22% correspond to the initial moisture content of three sample groups. In the context of the collapsibility test, these samples eventually reach a fully saturated state through moisture addition. Specifically, the choice of 18% aligns with the soil’s plastic limit moisture content, serving as a reference point. Accordingly, we have selected two groups of samples at 14% and 22% moisture content, representing comparatively drier and wetter states, respectively. Vertical loads are applied to these specimens under various water content conditions, and compression tests are performed until the vertical pressure reaches 400 kPa. The relationship between the deformation and stress of the samples is measured under each pressure level until these samples reach stability, that is when their deformation per hour is less than 0.01 mm. The average value of each group of samples is computed as the final test result to reduce the impact of soil heterogeneity.

Collapsibility test

• (1) The coefficient of collapse ${\delta }_{wp}$ is calculated as

  $${\delta }_{wp}=\frac{\Delta {h}_{2}}{h}\times 100\%=\frac{{h}_{p}-{h}_{p}{\prime}}{h}\times 100\%$$

  (1)

  where ${h}_{p}$ is the height of the specimen after stable deformation at a certain pressure level, ${h}_{p}{\prime}$ is the height of the sample after stable deformation due to immersion in water at a certain pressure level and $h$ is the initial height of the sample as shown in Fig. 3.

  According to the coefficient of collapsibility, collapsible loess can be divided into weak collapsibility (${\delta }_{wp}$< 0.03), medium collapsibility (${\delta }_{wp}$ = 0.03–0.07) and strong collapsibility (${\delta }_{wp}$ > 0.07) (Zhang et al. 2018).

• (2) The self-weight collapse coefficient is calculated as

  $${\delta }_{{\text{zs}}}=\frac{{h}_{z}-{h}_{z}{\prime}}{h}$$

  (2)

  where ${h}_{z}$ is the height of the sample after stable deformation under saturated self-weight pressure, and ${h}_{z}{\prime}$ is the height of the sample after stable deformation due to immersion and collapse under saturated self-weight pressure.

• (3) The relationship curve between pressure and collapse coefficient is drawn with pressure and collapse coefficient in the horizontal and vertical axes, respectively.

Fig. 3

Vertical deformation-pressure relationship

The collapsibility coefficient is gradually increased to 50, 100, 200 and 400 kPa. After reaching a stable compression deformation, the water content is increased, and the collapsibility coefficient is calculated at 50, 100, 200 and 400 kPa. When measuring the coefficient of self-weight collapse, the saturated self-weight pressures of the overlying soil at 2 m and 4 m are calculated at 85% saturation (34.74 kPa and 68.69 kPa, respectively). The overlying saturated self-weight soil pressure is gradually loaded, a water immersion test is performed to measure the amount of collapse and the self-weight collapse coefficient is computed.

Results and discussion

Compression test

Compression curve

As shown in Fig. 4, the difference between the compression curves decreases along with increasing initial water content, thereby indicating that the collapsibility of loess decreases along with increasing initial water content. Meanwhile, the compression curves for the un-immersed samples gradually curve from top to bottom along with an increasing initial water content. The compressibility coefficient of these un-immersed samples also significantly increases, thereby suggesting that the compressibility of loess increases and decreases along with moisture.

Fig. 4

Compression curves of remoulded loess

The e-lg p curves of un-immersed samples under different initial water contents in Fig. 5 show that the compressibility coefficient of these samples significantly increases along with initial water content, thereby indicating that the compressibility of loess increases and decreases along with water content.

Fig. 5

Compression curves of intact loess at a depth of 2 m

Relationship between compressive modulus and saturation

The plots depicted in Figs. 6, 7 and 8 exhibit a pronounced linear correlation. For the stress–strain relationship of unsaturated loess, the relationship between equivalent suction and saturation can be directly obtained using the principle of deformation equivalence and fitted using the exponential function. This relationship is in good agreement with the test results. Figures 6, 7 and 8 show a specific internal connection between the compression deformation and collapse deformation of loess. Specifically, compression deformation gradually increases with the humidification process, whereas collapse deformation gradually decreases along with increasing humidification level and eventually disappears. These results indicate that the pre-wetting of collapsible loess will transform collapsible deformation into loading deformation, that is, both the collapsibility degradation and compressibility increase. If the collapsible loess is dehumidified in advance, then the loading deformation will transform into collapsible deformation, that is, compressibility decreases, whilst collapsibility increases. The transformation of these two deformations during the humidification process is largely reversible. The compression modulus decreases along with increasing saturation, and the curve pattern is clear without obvious discrete points.

Fig. 6

The E_s-saturation relationship of remoulded loess at a depth of 2 m

Fig. 7

The E_s-saturation relationship of remoulded loess at a depth of 4 m

Fig. 8

The E_s-saturation relationship of intact loess at a depth of 2 m

Collapsibility test

After the solidified bond of loess is destroyed, such as that of collapsible loess soaked in water, its mechanical properties will change significantly. For instance, the compressibility of the loess increases, its bearing capacity decreases and collapsibility occurs. Although loess is formed under arid or semi-arid climatic conditions, this under-compacted soil does not have any load such that any loess soaked in water will experience collapse. Loess itself has a specific structural strength. When the loess has low pressure and is soaked in water, the shear stress generated at the contact point of its particles is smaller than its structural strength. Similar to clay soil, loess only produces a small amount of compression deformation. Only when the pressure increases to a specific value and the shear stress exceeds its structural strength will the deformation rate of loess suddenly accelerate.

Afterwards, the main physical and mechanical indicators of the undisturbed loess at 2 m and 4 m are measured by conducting specific gravity, particle analysis, permeability and liquid plastic limit tests as shown in Table 1.

Table 1 Basic physical parameters of loess

Collapsibility coefficient-saturation relationship of undisturbed loess

The corresponding pressure at the beginning of loess collapse is called the initial collapse pressure, which may also reflect the structural strength of loess after being soaked in water. When the actual pressure on the collapsible loess is equal to or greater than the initial pressure of soil collapse, the soil begins to collapse. On the contrary, if the pressure is less than the initial pressure of soil collapse, then compression deformation, rather than collapse deformation, occurs. Generally, the initial collapse pressure can be defined as the minimum collapse pressure corresponding to the collapse coefficient of collapsible loess (reaching up to 0.015). This indicator not only reflects the collapsibility of loess but also marks the beginning of loess collapse. With changes in pressure and humidity, the entire process curve of loess collapsibility consists of two turning points and three segments. Before the first turning point is the stage of structural strength compaction, between the two turning points is the stage of structural failure collapse, and the second turning point is the stage of consolidation formed by a new structure. The development of these three stages varies along with the applied pressure and humidification conditions.

The selection of the 9 drilled well locations was guided by the path taken by the Yumenkou Water Diversion East Expansion Project. Specifically, drilled 1 well is situated at Sanquan Reservoir, drilled 2 well at Beishe, drilled 3 well at Xichang Village, drilled 4 well at Longxiang Village, drilled 5 well at Xiji Village, drilled 6 well at Dongji Village, drilled 7 well at Beizhuang, drilled 8 well at Xibeiji, and drilled 9 well at Zheng Village. Figures 9 and 10 show that the collapsibility coefficient of undisturbed loess in (drilled 4 well) is the highest at 2 m or 4 m. The coefficient of collapse reaches its peak point with pressure variation, generally at around 200 kPa. When the pressure is low, the coefficient of collapse increases along with vertical pressure. When the vertical pressure is high, the coefficient of collapse decreases along with increasing pressure.

Fig. 9

The δs–p relationship of intact loess at a depth of 2 m

Fig. 10

The δs–p relationship of intact loess at a depth of 4 m

Effect of initial water content on loess collapsibility

Figure 11 shows the relationship between soil pressure and the collapsibility coefficient of remoulded soil samples taken from drilled 4 well at Yumenkou at depths of 2 m and 4 m. These figures show that the collapsibility coefficient gradually decreases along with increasing water content. At a low water content, the collapsibility coefficient significantly changes along with water content, whilst at a high water content, the change is not significant. Upon approaching saturation, the collapsibility of the loess disappears.

Fig. 11

The δs–p relationship of intact loess with different water contents

Analysis of loess collapsibility test

An essential external cause of loess collapse is the loss of suction in unsaturated loess after immersion in water. The role of suction in the structural strength of unsaturated loess cannot be ignored. When the water content is about 10%, the suction of loess can reach over 100 kPa, and when the water content is high, the suction can reach 400 kPa. At this point, suction has a significant contribution to the shear strength of the loess. Conversely, if the water content of unsaturated loess increases, the loss of suction will also result in a significant loss of loess shear strength. A greater loss of shear strength caused by the loss of suction during the collapse corresponds to a larger collapse coefficient.

The two sets of experiments show that the curve between the collapsibility coefficient of loess and the immersion pressure can be divided into four stages based on the deformation characteristics. The additional compression deformation stage generally shows a linear change, which indicates that after immersion, the shear stress caused by the external pressure between soil particles is less than the residual structural strength of the loess. At this point, the loess does not yet experience any damage to its intergranular arrangement structure and only shows a decrease in pore volume. At the initial collapse deformation stage, the external pressure exceeds the residual structural strength of the loess, and the connection between soil particles is destroyed. A slight increase in pressure can produce significant deformation, and the slope of the curve continues to increase. At the stage of significant development of collapsible deformation, the loess undergoes rapid collapsible deformation, and the slope of curve change reaches its maximum value. At the stage of collapsibility decline, due to the constraint effect of the ring knife sidewall, the loess cannot produce lateral extrusion, thereby reducing collapsibility and gradually increasing the curve deformation amplitude.

Improved K-G model and its application

Improved K-G model

Loess has all the properties of unsaturated soil. The surface tension at the interface between air and water in unsaturated soil increases the normal stress at the contact point of soil particles, thereby making the strength of unsaturated soil higher than that of saturated soil. Especially for collapsible loess, some significant differences can be observed in the deformation performance of unsaturated soils with different water contents.

How to reflect the influence of water content and its changes on the strength and deformation of unsaturated soil has always been a key issue in the study of unsaturated soil. Two physical quantities can reflect the internal moisture content of the soil, namely, suction and saturation. For specific unsaturated loess, the magnitude of suction mainly depends on the degree of saturation, whilst its relationship curve, commonly known as the soil water characteristic curve (SWCC), reflects the water-holding capacity of the soil.

Given the various shortcomings of previous models, K-G models with different water contents are used in this study to calculate the humidification deformation, which is simple and easy to measure, as shown in Eq. (3). The theory of these models is also relatively mature. The specific parameters of K-G models are shown in Table 2. Building formulas for matrix suction and saturation based on SWCCs can facilitate the computation of saturation in the numerical analysis.

$$\left\{\begin{array}{c}{K}_{t}={K}_{i}+{\alpha }_{k}\left(\frac{{\sigma }_{1}+2{\sigma }_{3}}{3}\right)\\ {G}_{t}={G}_{i}{\left[1-\frac{{R}_{f}\left({\sigma }_{1}-{\sigma }_{3}\right)\left(1-sin\varphi \right)}{2ccos\varphi +2{\sigma }_{3}sin\varphi }\right]}^{2}\end{array}\right.$$

(3)

where ${G}_{i}={Kp}_{a}{\left(\frac{{\sigma }_{3}}{{p}_{a}}\right)}^{n}$ is the initial shear modulus, ${R}_{f}=\frac{{\left({\sigma }_{1}-{\sigma }_{3}\right)}_{f}}{{\left({\sigma }_{1}-{\sigma }_{3}\right)}_{ult}}$ is the failure ratio, ${\left({\sigma }_{1}-{\sigma }_{3}\right)}_{f}=\frac{2ccos\varphi +2{\sigma }_{3}sin\varphi }{1-sin\varphi }$ is the Mohr–Coulomb yield criterion, c is the cohesion internal friction angle and $\varphi$ is the internal friction angle.

Table 2 K-G model parameters

The SWCC must be determined through tests. For convenience of analysis, the following hyperbola saturation and suction relationship are adopted in the subsequent tests:

$$s={s}_{0}\left({s}_{r}^{-\frac{1}{{m}_{1}}}-1\right)$$

(4)

where ${s}_{0}$ is the suction force when ${s}_{r}={2}^{-{m}_{1}}$, ${m}_{1}$ = 1.25 and ${s}_{0}$ = 75 kPa.

For unsaturated porous media, the permeability coefficient is a function of saturation (Sr) and volumetric water content (θ). Given that the relationship between volumetric water content and saturation and matrix suction can be reflected by the SWCC, the permeability coefficient is also a function of matrix suction. This article adopts the following relationship in the experiments (Zhan et al. 2023):

$$k={k}_{s}exp\left(-{c}_{k}\frac{s}{{p}_{a}}\right)$$

(5)

where ${k}_{s}$ is the permeability coefficient of saturated soil, ${c}_{k}$ is an empirical constant with a value of 0.1 and ${p}_{a}$ is the standard atmospheric pressure.

Table 2 shows that the various parameters of the K-G model vary along with water content. Specifically, along with increasing water content, K_i, α_k decreases, but the overall change is not significant, K, c decreases, and the amplitude of change is significant, φ, R_f does not change significantly but shows a decreasing trend, and n increases.

Numerical simulation

Basic equations for numerical experiments

K-G model uses the theory and method of bearing deformation of soil and still needs to follow the framework of standard research methods, that is, according to the dynamic balance equation of soil element and pore fluid, effective stress principle, seepage continuity equation, geometric equation and physical equation to establish the basic static equation of soil, and then conduct numerical analysis to obtain the stress, strain and pore pressure field of soil, furthermore evaluate the stability of unsaturated and saturated soils.

Governing equation

Equilibrium equation of soil mass

When describing the stress and strain of axisymmetric problems, using cylindrical coordinates is more convenient than using the Cartesian coordinate system. As shown in Fig. 12, taking the axis of symmetry of the elastic body as the z-axis, all stress, strain and displacement components are functions of r and z and do not change with θ. A small hexahedron is cut from the elastic body using two cylindrical surfaces separated by two perpendicular planes at angle dθ to each other and by two horizontal planes separated by dz.

Fig. 12

Stress balance diagram

The normal stresses in the r-, θ- and z-directions are called the normal radial, normal circumferential and normal axial stresses, which are denoted by ${\sigma }_{r}$, ${\sigma }_{\theta }$ and ${\sigma }_{{\text{z}}}$, respectively. Meanwhile, the shear stress acting in the r-direction on a plane perpendicular to the z-axis is denoted by ${\tau }_{zr}$, whilst that acting in the z-direction on a cylindrical surface is denoted by ${\tau }_{rz}$. According to the equivalent law of shearing stress, ${\tau }_{zr}={\tau }_{rz}$ as denoted by ${\tau }_{zr}$. According to the symmetry condition, the remaining shear stress components ${\tau }_{r\theta }={\tau }_{\theta r}$ and ${\tau }_{\theta z}={\tau }_{z\theta }$ do not exist. Therefore, only the four stress components ${\sigma }_{r},{\sigma }_{\theta },{\sigma }_{z},{\tau }_{zr}$ need to be considered, and their overall values are represented by $\left\{\sigma \right\}$ as shown in Eq. (6).

$$\left\{\sigma \right\}=\left\{\begin{array}{c}{\sigma }_{r}\\ {\sigma }_{\theta }\\ {\sigma }_{z}\\ {\tau }_{zr}\end{array}\right\}$$

(6)

Corresponding to the four stress components mentioned above are four strain components. The normal strains along the r-, θ- and z-directions are called the normal radial, normal circumferential and normal axial strains, which are denoted by ${\varepsilon }_{r}$, ${\varepsilon }_{\theta }$ and ${\varepsilon }_{z}$, respectively. Meanwhile, the shear strain between the r- and z-directions is represented by ${\gamma }_{zr}$. Due to the symmetry condition, the strain components ${\gamma }_{r\theta }$ and ${\gamma }_{\theta z}$ are both 0. The total of these four strain components is represented by $\left\{\varepsilon \right\}$ as shown in Eq. (7).

$$\left\{\varepsilon \right\}=\left\{\begin{array}{c}{\varepsilon }_{r}\\ {\varepsilon }_{\theta }\\ {\varepsilon }_{z}\\ {\gamma }_{zr}\end{array}\right\}$$

(7)

The equilibrium differential equation for axisymmetric problems can be expressed as

$$\left.\begin{array}{c}\frac{\partial {\sigma }_{\text{r}}}{\partial {\text{r}}}+\frac{\partial {\tau }_{\text{rz}}}{\partial {\text{z}}}+\frac{{\sigma }_{\text{r}}-{\sigma }_{\theta }}{\text{r}}={0}\\ \frac{\partial {\tau }_{\text{rz}}}{\partial {\text{r}}}+\frac{\partial {\sigma }_{\text{z}}}{\partial {\text{z}}}+\frac{{\tau }_{\text{rz}}}{\text{r}}=\gamma \end{array}\right\}$$

(8)

Considering the nonlinearity of deformation, the equilibrium equation is written in the following incremental form:

$${\left[L\right]}^{T}\left\{\Delta \sigma \right\}\text{=}\left\{\Delta F\right\}$$

(9)

where $\left\{\Delta F\right\}$ is the increment of body force, and ${\left[L\right]}^{T}$ is the balanced differential operator computed as

$${\left[L\right]}^{T}=\left[\begin{array}{cccccc}\frac{\partial }{\partial x}& 0& 0& \frac{\partial }{\partial y}& 0& 0\\ 0& \frac{\partial }{\partial y}& 0& \frac{\partial }{\partial x}& \frac{\partial }{\partial z}& 0\\ 0& 0& \frac{\partial }{\partial c}& 0& \frac{\partial }{\partial y}& \frac{\partial }{\partial z}\end{array}\right]$$

(10)

Deformation equation of soil skeleton

$$\left\{\Delta \varepsilon \right\}=-\left[L\right]\left\{\Delta u\right\}$$

(11)

where $\left\{\Delta u\right\}$ is the displacement increment.

Constitutive equation of soil skeleton

Considering the nonlinearity of the deformation, the constitutive equation of the soil skeleton is expressed in incremental form as follows:

$$\left\{\Delta \sigma {\prime}\right\}={\left[D\right]}_{t}\left\{\Delta \varepsilon \right\}$$

(12)

where ${\left[D\right]}_{t}$ is the tangent modulus matrix of the stress–strain relationship of the soil skeleton.

Effective stress principle

Under static conditions, the effective stress principle is expressed as total stress ${\tilde{\sigma }}_{i}$(${\tilde{\sigma }}_{x}$,${\tilde{\sigma }}_{y}$,${\tilde{\sigma }}_{z}$), which is equal to the sum of the effective stress borne by the soil skeleton ${\sigma }_{i}$ (${\sigma }_{x}$,${\sigma }_{y}$,${\sigma }_{z}$) and the pore water pressure ${u}_{w}$:

$$\left.\begin{array}{c}{\tilde{\sigma }}_{x}={\sigma }_{x}+{u}_{w}\\ {\tilde{\sigma }}_{y}={\sigma }_{y}+{u}_{w}\\ {\tilde{\sigma }}_{r}={\sigma }_{r}+{u}_{w}\end{array}\right\}$$

(13)

Geometrical equation

The mathematical expression describing the relationship between the strain and displacement components is called a geometric equation. According to the assumption of minor strain and the assumption that strain is positive in compression, the geometric equation of soil is expressed as

$$\begin{array}{cc}{\varepsilon }_{x}=-\frac{\partial u}{\partial x}& ,{\gamma }_{xy}=-\left(\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\right)\\ {\varepsilon }_{y}=-\frac{\partial v}{\partial y}& ,{\gamma }_{yz}=-\left(\frac{\partial v}{\partial r}+\frac{\partial w}{\partial y}\right)\\ {\varepsilon }_{r}=-\frac{\partial w}{\partial r}& ,{\gamma }_{rx}=-\left(\frac{\partial w}{\partial x}+\frac{\partial u}{\partial r}\right)\end{array}$$

(14)

Pore fluid equilibrium equation

When establishing the fluid balance equation, the soil mass is studied according to the seepage model in hydraulics. When determining the seepage velocity and permeability coefficient, the entire space in the seepage area is filled with fluid, there is no soil skeleton, and only the resistance of the soil skeleton to the seepage movement is considered.

Take for example a microelement in the seepage zone (Fig. 13) that is filled with fluid. The fluid accompanies the soil skeleton moving with accelerations $\ddot{u}$ and $\ddot{v}$. At the same time, the fluid undergoes seepage motion relative to the soil skeleton with flow velocities ${q}_{x}$ and ${q}_{z}$. The stresses acting on the microelement include pore pressure p, seepage resistance ${f}_{x}={f}_{z}={\rho }_{w}{g}_{i}$ exerted by the soil skeleton on the pore water, volumetric force ${\rho }_{w}g$ caused by gravity (where ${\rho }_{w}$ is the density of water), seepage inertia force (${\rho }_{w}{\dot{q}}_{x}$, ${\rho }_{w}{\dot{q}}_{z}$) and the inertia force caused by the movement of the soil skeleton ($\rho \ddot{u}$, $\rho \ddot{v}$). The inertia forces caused by the seepage and soil skeleton movement are ignored. The pore fluid equilibrium equation conforms to Darcy’s law as shown in Eq. (15).

$$\left.\begin{array}{c}{q}_{x}=-\frac{{k}_{x}}{{\gamma }_{w}}\frac{\partial {u}_{w}}{\partial x}={\overline{k} }_{x}\frac{\partial {u}_{w}}{\partial x}\\ {q}_{Z}=-\frac{{k}_{y}}{{\gamma }_{w}}\frac{\partial {u}_{w}}{\partial z}={\overline{k} }_{y}\frac{\partial {u}_{w}}{\partial z}\end{array}\right\}$$

(15)

where ${k}_{x}$ and ${k}_{z}$ are the permeability coefficients in the x- and z-directions, respectively, and ${q}_{x}$ and ${q}_{z}$ are the seepage velocities in the x- and z-directions.

Fig. 13

Fluid balance diagram within pores

Initial and boundary conditions

The consolidation problem of saturated or unsaturated soil should be solved according to the boundary value problem, and the initial and boundary conditions serve as the basic definite solution conditions of this problem. To obtain a correct and unique solution, considering the initial and boundary conditions is essential.

Initial condition

For the consolidation problem of saturated soil, the initial conditions are clear and easy to determine, and the calculation time domain is divided into several time steps $\Delta t$. Starting from t = 0, the finite element equation system is solved once for every additional $\Delta t$. The load, displacement and pore pressure must be known at time t = 0 to calculate the changes in displacement and pore pressure for the next period.

In the consolidation theory of unsaturated soil, temperature and air pressure are not considered essential field variables, but the following considerations should still be given when assigning initial values:

• (a) When the temperature changes are ignored, the temperature within the analysis range is uniformly distributed and stable. In the analysis, the temperature within the analysis range may be regarded as the average temperature around the analysis range at the time and remains unchanged.

• (b) In most problems, the initial state is usually the stable state of the soil that has lasted for a long time before loading. Therefore, the initial pore air pressure in the soil can be 0.

For the essential field variables in the consolidation theory of unsaturated soil, initial values should be given according to the situation, but this problem needs to be studied further using Eq. (16).

$$\left\{\begin{array}{c}u\\ \varepsilon \\ {u}_{w}\\ \sigma \\ n\end{array}\right\}=\left\{\begin{array}{c}{u}_{0}\\ {\varepsilon }_{0}\\ {u}_{w}\\ {\sigma }_{0}\\ {n}_{0}\end{array}\right\},\left(x,y,z\in \Omega ,t=0\right)$$

(16)

where $\Omega$ represents the analysis range, and $u$, $\varepsilon$, ${u}_{w}$, $\sigma$ and $n$ represent displacement, strain, pore water pressure, stress and pore ratio, respectively.

For static problems, the state before loading is usually used as a reference. Therefore, the initial displacement and initial strain are usually set as 0, and the initial pore water pressure, stress field and pore ratio should be determined using the values in the soil before loading, which, in turn, can be obtained using iterative methods.

Boundary condition

For unsaturated soil, the boundary conditions only need to deal with boundary problems related to the four essential variables. These variables can be divided into two categories, namely, displacement and pore water pressure. Each of these categories corresponds to two boundary conditions, namely, natural and forced boundary conditions. These boundary conditions are expressed in increments as shown in Eqs. (17) to (19).

$$\begin{array}{cc}\left[\overline M\right]\;\left\{\triangle\sigma'\right\}=\left\{\triangle\overline F\right\},\left(x,\;y,\;z\right)\in\mathrm\Gamma_n^1,\;t>0&\left(\mathrm{Load}\;\mathrm{boundary}\right)\end{array}$$

(17)

$$\begin{array}{cc}\left\{\Delta{\text{u}}_\text{w}\right\}=\left\{\Delta\text{u}_\text{w}^-\right\},\left(x,y,z\right)\in\Gamma_c^2,t>0&(\mathrm{Water}\;\mathrm{head}\;\mathrm{boundary})\end{array}$$

(18)

$$\begin{array}{cc}\Delta {\overline{q} }_{w}\cdot \overline{N }=\Delta {\overline{q} }_{w}, \left(x,y,z\right)\in {\Gamma }_{n}^{2},t>0& (\mathrm{Flow\ boundary})\end{array}$$

(19)

where $\overline{N }$ is the outer normal vector of the boundary interface element, $\overline{N }={\left(l,m,n\right)}^{T}$, and $\left[\overline{M }\right]=\left[\begin{array}{cccccc}{l}{\prime}& 0& 0& {m}{\prime}& 0& {n}{\prime}\\ 0& {m}{\prime}& 0& {l}{\prime}& {n}{\prime}& 0\\ 0& 0& {n}{\prime}& 0& {m}{\prime}& {l}{\prime}\end{array}\right]$, ${l}{\prime}={\text{cos}}(\overrightarrow{N},x)$, ${m}{\prime}={\text{cos}}(\overrightarrow{N},y)$, ${n}{\prime}={\text{cos}}(\overrightarrow{N},z)$.

${\Gamma }^{\alpha }$ is the outer boundary of the analysis area, ${\Gamma }_{\text{n}}^{\alpha }$ is the boundary where the natural boundary conditions are located and ${\Gamma }_{\text{c}}^{\alpha }$ is the boundary where the mandatory boundary conditions are located ($\alpha$ = 1, 2) that satisfies the following conditions: ${\Gamma }_{n}^{\alpha }\cup {\Gamma }_{c}^{\alpha }={\Gamma }^{\alpha }$, ${\Gamma }_{\text{n}}^{\alpha }\cap {\Gamma }_{\text{C}}^{\alpha }=\Phi$, ($\alpha$ = 1,2).

The pore pressure of the permeable boundary is known, and continuity equations do not need to be established for these nodes. The impermeable boundary needs to establish a continuity equation, but the boundary flow rate is 0, and these points are not different from the internal points. For the problem of foundation consolidation, due to the infinite size of the foundation, the actual calculation area is limited. In this case, for the truncated boundary, the pore pressure can be determined using the extrapolation method.

Determination of $\Delta t$

In the finite element consolidation calculation, attention should be paid to the selection of time step $\Delta t$. A very small $\Delta t$ can lead to calculation instability and distortion. However, at the beginning of consolidation, a very large $\Delta t$ can lead to significant calculation errors. Mikheev (2000) proposed the following equation for estimating time steps:

$$\Delta t=\frac{{L}^{2}}{\frac{k}{{\gamma }_{w}}\left(K+\frac{4}{3}G\right)}$$

(20)

where L is the size in the direction of unit drainage, k is the permeability coefficient, K is the bulk modulus of the soil and G is the shear modulus of the soil.

Results of numerical simulation

The immersion pit test can be simplified as a plane strain problem, and the finite element equation in the form of plane strain can be obtained through the transformation of the axisymmetric equation. In the calculation, the displacement is interpolated using a quadrilateral eight-node isoparametric element, whilst the seepage is calculated using a quadrilateral four-node isoparametric element. The immersion pit has a circular shape with a radius of 5 m, which conforms to the axisymmetric model. A half-symmetric model is then constructed with an immersion width of 5 m, outer width of 11 m and depth of 13.9 m. The immersion amount is calculated based on an average of 65 m³ per day, and a water head of 30 cm is added to the infiltration boundary. The calculation model is shown in Fig. 14.

Fig. 14

Computing model: a schematic diagram of immersion location and b finite element meshes

The numerical calculation parameters are determined based on the K-G model test results for undisturbed loess. These parameters are defined as follows: K = 96.338 kPa, n = 0.211, R_f = 0.861, c = 22 kPa, φ = 30.82°, K_i = 1.545 MPa and α_k = 4.134. When calculating the initial stress field, the bulk density of the soil is 13.8 × 10³ kN/m³. In order to reflect the real-time changes in model parameters with saturation, a function of the variation of model parameters with saturation is developed in the laboratory as shown in Eq. (21).

$$\begin{array}{c}K=-43.952ln({s}_{r})+236.46\\ n=0.4788ln({s}_{r})-1.3148\\ {R}_{f}=-0.1213ln({s}_{r})+1.2165\\ c=-8.8893ln({s}_{r})+50.808\\ \phi =-1.9371ln({s}_{r})+36.518\\ {K}_{i}=-0.3193ln({s}_{r})+2.5311\end{array}$$

(21)

The function of each parameter changing with saturation is introduced into the finite element calculation to reflect the influence of saturation on the soil modulus. The distributions of displacement, stress and saturation at different times are shown in Figs. 15, 16 and 17, whereas the variation in the simulated maximum vertical displacement with the number of days of immersion is shown in Fig. 18.

Fig. 15

Soak in water for 5 days: a shear stress distribution field; b horizontal displacement distribution field; c vertical displacement distribution field; and d saturation distribution field

Fig. 16

Soak in water for 30 days: a shear stress distribution field; b horizontal displacement distribution field; c vertical displacement distribution field; and d saturation distribution field

Fig. 17

Soak in water for 60 days: a shear stress distribution field; b horizontal displacement distribution field; c vertical displacement distribution field; and d saturation distribution field

Fig. 18

Maximum settlement for different immersion days

Model validation

Field immersion test

The infiltration of water will change the negative pore water pressure distribution of the foundation within a short period, thereby leading to the deformation of the loess foundation. The Donglintuo Village large-scale trial immersion pit test (Fig. 19) is selected as an example in this study to conduct a coupled analysis of seepage deformation on uneven settlements caused by infiltration in a loess site in order to understand how the foundation deformation field and saturation change along with immersion time. The on-site test is conducted for 60 days, whilst the immersion test is conducted for 46 days. After the immersion reaches stability (the average collapse in the last 5 days is less than 1 mm/d), the immersion in the test pit is terminated, and an observation is conducted for 14 days after the water stops. In the early immersion stage, the daily water consumption is unstable and belongs to an unsteady process. The test process is shown in Table 3. As the soil in the test pit gradually saturates, the water consumption tends to stabilise with a stable daily water consumption of nearly 65 m³. The total water consumption during the soaking process is 2989.5 m³ (considering the effects of evaporation and rainfall processes). In order to simulate the actual on-site immersion as closely as possible, the numerical experiment also simulates the 60-day immersion, and the immersion amount is analysed based on stable daily water consumption. The test results are shown in Fig. 20.

Fig. 19

Layout of monitoring points for the field immersion test in Donglintuo Village: a field pictures and b layout of monitoring points

Table 3 Settlement data of the immersion pit test

Fig. 20

Measured ground settlement map: a section A-A and b section B-B

Comparative analysis

• (1) The soil cannot easily reach saturation below the pit during the simulation of the immersion test pit. The saturation field gradually decreases from the centre to the edge of the pit, thereby indicating that the closer it is to the edge of the test pit, the harder it is for the soil to reach saturation. The changes in the saturation field indicate that water migrates downwards over time. The ratio of horizontal displacement to vertical displacement is 0.067, which conforms to the law of soil deformation. The change in the shear stress field is not very obvious, thereby indicating that the influence of immersion on shear stress is minimal.

• (2) The maximum vertical displacement of the test pit is 0.036 m, and the most severe change is recorded after 20 days. The vertical displacement reaches 0.018 m, completing half of the total vertical displacement.

• (3) Compared with that recorded in the immersion pit test, the numerical model shows a 12.5% increase in vertical displacement, thereby indicating that the numerical simulation is in line with the immersion pit test.

Conclusion and recommendation

The parameters of the improved K-G model are obtained by conducting indoor compression and collapse tests, and the deformation characteristics of loess collapse are studied through a numerical simulation and a large-scale trial immersion pit test. The following conclusions are obtained:

• (1) The collapsibility of loess decreases along with increasing initial water content and depth. When the water content is high, the e-lg p curves under different initial water contents tend to overlap.

• (2) The maximum loess collapse coefficient is obtained at a pressure of around 200 kPa. According to the pressure and collapse coefficient relationship curve, the collapse deformation of loess can be divided into four stages, namely, the additional compression deformation stage, initial collapse deformation stage, significant development stage of collapse deformation and collapse decline change stage.

• (3) A calculation model that considers the unsaturated collapsible deformation of loess is proposed by taking saturation as a state variable and by combining unsaturated seepage and unsaturated deformation theories. This method can obtain the variation in collapsible deformation along with water infiltration time. The model parameters are simply defined by performing several triaxial tests with different initial water contents on a conventional triaxial instrument. When analysing practical engineering problems, numerical simulation analysis can be used to calculate the collapsible deformation of unsaturated loess under humidification conditions, thereby avoiding the measurement and calculation of suction.

• (4) The comparison between the calculated results of the improved K-G model and the large-scale trial immersion pit test shows that more practical results can be obtained when the unsaturated collapse deformation of loess is considered.